Approximating Node-Weighted k-MST on Planar Graphs
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We study the problem of finding a minimum weight connected subgraph spanning at least k vertices on planar, node-weighted graphs. We give a (4+)-approximation algorithm for this problem. In the process, we use the recent LMP primal-dual 3-approximation for the node-weighted prize-collecting Steiner tree problem and the Lagrangian relaxation. In particular, we improve the procedure of picking additional vertices given by Sadeghian by taking a constant number of recursive steps and utilizing the limited guessing procedure of Arora and Karakostats. We argue that our approach can be interpreted as a generalization of a result by Könemann et al. Together with a result by Mestre this implies that our bound is essentially best possible among algorithms that utilize an LMP algorithm for the Lagrangian relaxation as a black box.
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